On submanifolds satisfying chen’s equality in a real space form

Abstract EN

Einstein, conformally flat, semisymmetric, and Ricci-semisymmetric submanifolds satisfying Chen’s equality in a real space form are studied.

We prove that an n-dimensional (n ≥ 3) submanifold of a real space form M~ n + m (c) satisfying Chen’s equality is (i) Einstein if and only if it is a totally geodesic submanifold of constant curvature c ; and (ii) conformally flat if and only if inf K = c, where K denotes the sectional curvatures of the submanifold.

We also classify semisymmetric and Riccisemisymmetric submanifolds satisfying Chen’s equality in a real space form. 2000 Mathematics Subject Classification : 53C40, 53C25, 53C42.